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Chapter 7 Practice Exercises 547
Chapter 7 Practice Exercises
Differentiation
In Exercises 1–24, find the derivative of y with respect to the appropri-
ate variable.
1. 2.
3. 4.
5. 6.
7. 8.
9. 10.
11. 12.
13. 14.
15.
16. 17.
18.
19.
20. y = s1 + t2d cot-1 2t
y = t tan-1 t - 1
2
 ln t
y = z cos-1 z - 21 - z2 y = ln cos-1 xy = sin-1 a 12y b , y 7 1
y = sin-121 - u2, 0 6 u 6 1y = 2sln xdx>2y = sx + 2dx + 2 y = 22x-22y = 5x3.6
y = 92ty = 8-t
y = log5 s3x - 7dy = log2 sx2>2d
y = ln ssec2 udy = ln ssin2 ud
y = x2e-2>xy = 1
4
 xe4x - 1
16
 e4x
y = 22e22xy = 10e-x>5
21.
22.
23.
24.
Logarithmic Differentiation
In Exercises 25–30, use logarithmic differentiation to find the deriva-
tive of y with respect to the appropriate variable.
25. 26.
27.
28.
29. 30.
Integration
Evaluate the integrals in Exercises 31–78.
31. 32. Le
t cos s3et - 2d dtLe
x sin sexd dx
y = sln xd1>sln xdy = ssin ud2uy =
2u2u2u2 + 1y = a
st + 1dst - 1d
st - 2dst + 3d b
5
, t 7 2
y = 10A3x + 42x - 4y = 2sx2 + 1d2cos 2x
y = s1 + x2de tan
-1 x
y = csc-1 ssec ud, 0 6 u 6 p>2
y = 22x - 1 sec-11xy = z sec-1 z - 2z2 - 1, z 7 1
4100 AWL/Thomas_ch07p466-552 8/20/04 10:03 AM Page 547
33.
34.
35. 36.
37. 38.
39. 40.
41. 42.
43. 44.
45. 46.
47. 48.
49. 50.
51. 52.
53. 54.
55. 56.
57. 58.
59. 60.
61. 62.
63. 64.
65. 66.
67. 68.
69. 70.
71. 72.
73. 74. L 
dx2-x2 + 4x - 1L dx2-2x - x2 L
-26>25
-2>25 dyƒ y ƒ25y2 - 3L2>322>3 dyƒ y ƒ29y2 - 1 L
 
24 dy
y2y2 - 16L dyy24y2 - 1 L
323 dt3 + t2L2-2 3 dt4 + 3t2
L
1>5
-1>5
 
6 dx24 - 25x2L3>4-3>4 6 dx29 - 4x2 L
e
1
 
8 ln 3 log3 u
u
 duL
8
1
 
log4 u
u
 du
L
4
2
s1 + ln tdt ln t dtL
3
1
 
sln sy + 1dd2
y + 1 dy
L
e2
e
 
1
x2ln x dxL e1 1x s1 + 7 ln xd-1>3 dx L
ln 9
0
 euseu - 1d1>2 duL
ln 5
0
 ers3er + 1d-3>2 dr
L
0
-ln 2
 e2w dwL
-1
-2
 e-sx + 1d dx
L
8
1
 a 2
3x
- 8
x2
b dxL
4
1
 ax
8
+ 1
2x
b dx
L
32
1
 
1
5x
 dxL
7
1
 
3
x dx
L 2
tan x sec2 x dxLx3
x2 dx
L 
cos s1 - ln yd
y dyL 
1
r csc
2 s1 + ln rd dr
L 
ln sx - 5d
x - 5 dxL 
sln xd-3
x dx
L 
dy
y ln yL 
tan sln yd
y dy
L
p>6
-p>2
 
cos t
1 - sin t dtL
4
0
 
2t
t2 - 25
 dt
L
1>4
1>6
 2 cot px dxL
p
0
 tan 
x
3
 dx
L
e
1
 
2ln x
x dxL
1
-1
 
dx
3x - 4
L csc
2 x ecot x dxL sec
2 sxde tan x dx
Le
y csc sey + 1d cot sey + 1d dy
Le
x sec2 sex - 7d dx 75. 76.
77. 78.
Solving Equations with Logarithmic
or Exponential Terms
In Exercises 79–84, solve for y.
79. 80.
81. 82.
83. 84.
Evaluating Limits
Find the limits in Exercises 85–96.
85. 86.
87. 88.
89. 90.
91. 92.
93. 94.
95. 96.
Comparing Growth Rates of Functions
97. Does ƒ grow faster, slower, or at the same rate as g as 
Give reasons for your answers.
a.
b.
c.
d.
e.
f.
98. Does ƒ grow faster, slower, or at the same rate as g as 
Give reasons for your answers.
a.
b.
c.
d.
e.
f. gsxd = e-xƒsxd = sech x,
gsxd = 1>x2ƒsxd = sin-1s1>xd,
gsxd = 1>xƒsxd = tan-1s1>xd,
gsxd = exƒsxd = 10x3 + 2x2,
gsxd = ln x2ƒsxd = ln 2x,
gsxd = 2-xƒsxd = 3-x,
x : q ?
gsxd = exƒsxd = sinh x,
gsxd = 1>xƒsxd = csc-1 x,
gsxd = tan-1 xƒsxd = x,
gsxd = xe-xƒsxd = x>100,
gsxd = x + 1xƒsxd = x,
gsxd = log3 xƒsxd = log2 x,
x : q ?
lim
x:0+ a1 +
3
x b
x
lim
x: q a1 +
3
x b
x
lim
y:0+ e
-1>y ln ylim
t:0+ a
et
t -
1
t b
lim
x:4 
sin2 spxd
ex - 4 + 3 - x
lim
t:0+ 
t - ln s1 + 2td
t2
lim
x:0 
4 - 4ex
xex
lim
x:0 
5 - 5 cos x
ex - x - 1
lim
x:0 
2-sin x - 1
ex - 1limx:0 
2sin x - 1
ex - 1
lim
u:0 
3u - 1
u
lim
x:0 
10x - 1
x
ln s10 ln yd = ln 5xln sy - 1d = x + ln y
3y = 3 ln x9e2y = x2
4-y = 3y + 23y = 2y + 1
L 
dt
s3t + 1d29t2 + 6tL dtst + 1d2t2 + 2t - 8 L
1
-1
 
3 dy
4y2 + 4y + 4L
-1
-2
 
2 dy
y2 + 4y + 5
548 Chapter 7: Transcendental Functions
4100 AWL/Thomas_ch07p466-552 8/20/04 10:03 AM Page 548
Chapter 7 Practice Exercises 549
99. True, or false? Give reasons for your answers.
a. b.
c. d.
e. f.
100. True, or false? Give reasons for your answers.
a. b.
c. d.
e. f.
Theory and Applications
101. The function being differentiable and one-to-
one, has a differentiable inverse Find the value of
at the point ƒ(ln 2).
102. Find the inverse of the function Then
show that and that
In Exercises 103 and 104, find the absolute maximum and minimum
values of each function on the given interval.
103.
104.
105. Area Find the area between the curve and the x-
axis from to 
106. Area
a. Show that the area between the curve and the x-axis
from to is the same as the area between the
curve and the x-axis from to 
b. Show that the area between the curve and the x-axis
from ka to kb is the same as the area between the curve and
the x-axis from to 
107. A particle is traveling upward and to the right along the curve
Its x-coordinate is increasing at the rate 
At what rate is the y-coordinate changing at the point
108. A girl is sliding down a slide shaped like the curve 
Her y-coordinate is changing at the rate 
At approximately what rate is her x-coordinate changing
when she reaches the bottom of the slide at (Take to
be 20 and round your answer to the nearest ft sec.)
109. The rectangle shown here has one side on the positive y-axis,
one side on the positive x-axis, and its upper right-hand vertex
on the curve What dimensions give the rectangle its
largest area, and what is that area?
y = e-x
2
.
> e
3x = 9 ft?
ft>sec .
s -1>4d29 - ydy>dt = y = 9e-x>3 .se2, 2d?1x m>sec .
sdx>dtd =y = ln x .
x = b s0 6 a 6 b, k 7 0d .x = a
y = 1>x
x = 2.x = 1
x = 20x = 10
y = 1>x
x = e .x = 1
y = 2sln xd/x
y = 10xs2 - ln xd, s0, e2]
y = x ln 2x - x, c 1
2e
, 
e
2
d
dƒ -1
dx
 `
ƒsxd
= 1
ƒ¿sxd .
ƒ -1sƒsxdd = ƒsƒ -1sxdd = x
ƒsxd = 1 + s1>xd, x Z 0.
dƒ -1>dx
ƒ -1sxd .
ƒsxd = ex + x ,
sinh x = Osexdsec-1 x = Os1d
ln 2x = Osln xdln x = osx + 1d
1
x4
= o a 1
x2
+ 1
x4
b1
x4
= O a 1
x2
+ 1
x4
b
cosh x = Osexdtan-1 x = Os1d
ln sln xd = osln xdx = osx + ln xd
1
x2
+ 1
x4
= O a 1
x4
b1
x2
+ 1
x4
= O a 1
x2
b
110. The rectangle shown here has one side on the positive y-axis,
one side on the positive x-axis, and its upper right-hand vertex
on the curve What dimensions give the rectangle
its largest area, and what is that area?
111. The functions and differ by a con-
stant. What constant? Give reasons for your answer.
112. a. If must 
b. If must 
Give reasons for your answers.
113. The quotient has a constant value. What value?
Give reasons for your answer.
114. vs. How does compare with
Here is one way to find out.
a. Use the equation to express ƒ(x) and
g(x) in terms of natural logarithms.
b. Graph ƒ and g together. Comment on the behavior of ƒ in re-
lation to the signs and values of g.
115. Graph the following functions and use what you see to locate
and estimate the extreme values, identify the coordinates of the
inflection points, and identify the intervals on which the graphs
are concave up and concave down. Then confirm your estimates
by working with the functions’ derivatives.
a. b. c.
116. Graph Does the function appear to have an ab-
solute minimum value? Confirm your answer with calculus.
117. What is the age of a sample of charcoal in which 90% of the car-
bon-14 originally present has decayed?
118. Cooling a pie A deep-dish apple pie, whose internal tempera-
turewas 220°F when removed from the oven, was set out on a
breezy 40°F porch to cool. Fifteen minutes later, the pie’s inter-
nal temperature was 180°F. How long did it take the pie to cool
from there to 70°F?
119. Locating a solar station You are under contract to build a so-
lar station at ground level on the east–west line between the two
buildings shown here. How far from the taller building should
you place the station to maximize the number of hours it will be
ƒsxd = x ln x .
y = s1 + xde-xy = e-x
2
y = sln xd>1x
loga b = sln bd>sln ad
gsxd = log2 sxd?
ƒsxd = logx s2dlog2 sxdlogx s2d
slog4 xd>slog2 xd
x = 1>2?sln xd>x = -2 ln 2 ,
x = 2?sln xd>x = sln 2d>2,
gsxd = ln 3xƒsxd = ln 5x
x
y
0
0.2 y � 
1
0.1
x2
ln x
y = sln xd>x2 .
x
y
0
1 y � e–x2
T
T
T
4100 AWL/Thomas_ch07p466-552 8/20/04 10:03 AM Page 549
in the sun on a day when the sun passes directly overhead? Begin
by observing that
Then find the value of x that maximizes 
x
50 m0
60 m
�
30 m
x
u .
u = p - cot-1 x
60
- cot-1 50 - x
30
.
120. A round underwater transmission cable consists of a core of cop-
per wires surrounded by nonconducting insulation. If x denotes
the ratio of the radius of the core to the thickness of the insula-
tion, it is known that the speed of the transmission signal is given
by the equation If the radius of the core is 1 cm,
what insulation thickness h will allow the greatest transmission
speed?
Insulation
x � rh
h
r
Core
y = x2 ln s1>xd .
550 Chapter 7: Transcendental Functions
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